Taken very literally, not all students are created equal — especially in their math learning skills, say Texas A&M University researchers who have found that not fully understanding the "equal sign" in a math problem could be a key to why U.S. students underperform their peers from other countries in math.

"About 70 percent of middle grades students in the United States exhibit misconceptions, but nearly none of the international students in Korea and China have a misunderstanding about the equal sign, and Turkish students exhibited far less incidence of the misconception than the U.S. students," note Robert M. Capraro and Mary Capraro of the Department of Teaching, Learning, and Culture at Texas A&M.

They have been trying to evaluate the success of math education through students' interpretation of the equal sign. They have published several articles on this topic, with the most recent one published in the February 2010 issue of the journal *Psychological Reports*.

Students who exhibit the correct understanding of the equal sign show the greatest achievement in mathematics and persist in fields that require mathematics proficiency like engineering, according to their research.

"The equal sign is pervasive and fundamentally linked to mathematics from kindergarten through upper-level calculus," Robert M. Capraro says. "The idea of symbols that convey relative meaning, such as the equal sign and "less than" and "greater than" signs, is complex and they serve as a precursor to ideas of variables, which also require the same level of abstract thinking."

The problem is students memorize procedures without fully understanding the mathematics, he notes.

"Students who have learned to memorize symbols and who have a limited understanding of the equal sign will tend to solve problems such as 4+3+2=( )+2 by adding the numbers on the left, and placing it in the parentheses, then add those terms and create another equal sign with the new answer," he explains. "So the work would look like 4+3+2=(9)+2=11.

"This response has been called a running equal sign — similar to how a calculator might work when the numbers and equal sign are entered as they appear in the sentence," he explains. "However, this understanding is incorrect. The correct solution makes both sides equal. So the understanding should be 4+3+2=(7)+2. Now both sides of the equal sign equal 9."

One cause of the problem might be the textbooks, the research shows.

The Texas A&M researchers examined textbooks in China and the United States and found "Chinese textbooks provided the best examples for students and that even the best U.S. textbooks, those sponsored by the National Science Foundation, were lacking relational examples about the equal sign."

Parents and teachers can help the students. The two researchers suggest using mathematics manipulatives and encourage teachers "to read professional journals, become informed about the problem and modify their instruction."

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## gunslingor1

All that being said, what on earth are you all talking about?

"...4+3+2=( )+2...So the work would look like 4+3+2=(9)+2=11"

I understand what your trying to teach kids with this kind of math problem, to add in segments, but its totally the wrong way to do it.

The mathematical notation 4+3+2=( )+2 literally means 4+3+2=0+2 which would be an inequality. Mathematical notations should not be circumvented to teach a point, this is a copout. There is a perfectly good correct way to do it. And you wonder why children are confused? How can you expect them to fully understand the equals sign when your not even using correct mathematical statements. It only confuses them more, and myself an engineer. Here's how to teach the idea 4+3+2=(a+b)+2.

## ainsworth50

## gunslingor1

I've had these problems many times. While in college, I was in Cal II or diffEQ or something similar, handling complex math. May roomate was training to be a restaurant manager, so he took pre-algebra or something. He always would ask for assistance in solving simple equations. Well, most of the time I could not help him without an hour of study to learn the teachers bizarre affectation of a proper notation.

I mean, the freaking article is saying students aren't understand the equals sign for some STrAnGE ReAson, yet they're demonstrating they also do not understand the darn equals sign. I mean, come on, mathematically 4+3+2=( )+2 means nothing. Equations are supposed to be universal, you're not supposed to learn new notations for every simple problem you come across.

## dirk_bruere

4+3+2 = x+2

What is x?

## Richak28

doing, therefore the ones they are teaching

doesn't know what they should be doing !!!

## RobertKarlStonjek

It might be less confusing if, as others above have suggested, correct notation is used ie 4+3+2=x+2, Solve the equation for x ie 4+3+2-2=x therefore x=4+3=7

## jsa09

I agree the notation should read 4+3+2=x+2 solve for x.

empty parenthesis is just asking for trouble and you will be sure to get it.

## gunslingor1

If an engineer can't directly solve a simple arithmatical equation using known notation, how can you expect a student to. I mean, notations are standardized for a reason.

## Sazzle

Really Sorry, but the Engineer's post begged a sarcastic response. I'm expecting a big bite back now - Ouch! ;-)

## tkjtkj

well, thanks to earlier contributor, x = a+b , so, we have:

4+3+2=(x[=a+b])+2

simply.

## gunslingor1

I don't think its the brits fault, lol. But we do have a problem with math in the US. Not so much a problem for engineers, physics dudes and math guys; the problem is for the lower math levels, english/law/medical majors and such.

These teachers are screwing with mathematics to try and make it easier to teach simple math with the justification that non-math guys will never need the real notation.

The example giving in this article is just one of MANY affectations of true math.

I sincerely feel approaching teaching math in this fashion is a huge mistake. They are putting a brick wall in front of these guys, if they do ever decide to learn more math. Plus, it makes it impossible for anyone who knows math to actually help them.

## Blicker

## Sazzle

## Eric_B

## gunslingor1

I remember my highschool chemistry teacher said a few things that really angered me; I challenged him and got suspended for this. Here's what he said:

1. "Global warming is largely caused by gas excaping from your tank while your fueling your car, so put the cap back on as quickly as you can."

-LOL, rediculous!

2. "Stomach acid can eat through a steak in 8 seconds, so why can't it eat through your stomach? The reasons is because stomach acid is diluted"

-No, it's because of polypeptides!

I only had a few intelligent teachers, the rest were practically incompetent.

## Cyberguy

Obviously the ( ) is supposed to represent an empty space where a number is to go. This is used in the very early stages before a child learns about x or y as variables - about the age of 6-8. In textboxs it is sometimes written as an empty square or an underscore instead of brackets. It is not a "notation" as such, just a teaching stage where the child tries to fill in the gaps. X, y, and other formal notations come later.

## jsa09

Why?

Let us start on the correct foot and save a lot of trouble later. I got sick of Primary school teachers simplifying stuff to such a level that it did not make sense at all.

How are kids going to learn maths properly anyway? "()" means something so why use incorrect notation and then have to relearn correct notation later?

Solving a problem with a single unknown is something any 5 year old can do and they don't care if you call it "X".

## Nik_2213

Then you have the issues of 'infix' and 'post-fix' notation, which are not necessarily self-evident. The priority of +-*/() can also confuse. Context is all, and may be ambiguous...

Too often, these concepts are badly taught, as 'doing' rather than 'understanding': The latter may take several different approaches and many more examples before a concept 'clicks', may require individual tuition to find an approach that a struggling student grasps...

## gunslingor1

I think your ineffectively overengineering your teaching methods. You say "Obviously the ( ) is supposed to represent an empty space". How is that obvious? There is no instruction indicating it and these are not standard mathematical operations; I personnaly would have thought that ()=0 and would have given "not equal" as the answer. "Find ()" would have covered it, afterall it doesn't matter what you call a variable, you could even call a variable "~&^#%~poopy". That would have been fine. But, I've seen in these courses, you often just present the problem (especially on tests) without explaining it. The result is a guy like me who can ace every test without studying, skipping classes because I already know the math, but still failing the test because your teaching in a very confusing manner.

You need to instill the proper mathematical techniques and principles from the begging, especially "listing all assumptions".